I've been in Boston for a bit over a year so I don't know it that well, but I've got a little bit of a clue.
First, if your into the college student scene, look into Allston (I spent my first year in Boston there), its got a rep as a dirty hipster college student nabe, but it's super fun and can be affordable esp. if you're open to roommates. If your into a little quieter scene look into Brookline, its just south of Allston and is still on the affordable side, but a bit more family friendly. Both of these neighborhoods are probably about a 30 min ride from Newbury St. on the green line.
Fenway is a bit closer, but a bit overpriced in my opinion. I've got as few friends who lived there and loved, but some find it to be a bit dead in the winter.
I'm in the South End, which is an easy walk up to Newbury and downtown, but a bit on the pricy side.
Definitely look into Cambridge / Sommerville. Both Porter and Davis Squares are super fun and have pretty easy red line access to the city. Places out there are typically a bit cheaper, but a lot are kinda crappy older houses.
Shoot me a PM if you have any specific questions or anything.
Best of luck!